EDB — 1KM

Exercises

1. [1KM]Let \(b∈ℝ\), \(n∈ℕ\). Assuming that \(f(t)=∑_{k=0}^∞ a_ k t^ k\) with radius of convergence \(r\) positive and \(t∈(-r,r)\), determine the coefficients \(a_ k\) so as to satisfy the following differential equations.,

   1. \(f'(t)=f(t)\) and \(f(0)=b\),

   2. \(f'(t)=t^ 2 f(t)\) and \(f(0)=b\),

   3. \(f''(t)=t^ 2 f(t)\) and \(f(0)=b,f'(0)=0\),

   4. \(t f''(t) + f'(t) + t f(t)=0\) and \(f(0)=b,f'(0)=0\),

   5. \(t^ 2 f''(t) + t f'(t) + (t^ 2-m^ 2)f(t)=0\) \(m≥ 2\) integer, \(f(0)=f'(0)=\ldots f^{(m-1)}=0\), and \(f^{(m)}=b\).

   (The last two are called Bessel equations). [[1KN]↺↻]

   Solution 1

   [1KP]↺↻